Physica 1llA (1982) 577-590 North-Holland Publishing Co.

MEMORY EFFECTS IN THE DIFFUSION OF AN

INTERACTING POLYDISPERSE SUSPENSION

II. HARD SPHERES AT LOW DENSITY

R.B. JONES and G.S. BURFIELD

Department of Physics, Queen Mary College. Mile End Road, London El 4N.S. Englund

Received 17 August 1981

We consider the diffusion of a low density suspension of polydisperse hard spheres. In a

previous article (I) we have obtained at long wavelength a general expression to first order in

density for the memory matrix appropriate to such a system. In the present article we evaluate

the memory rnatrix in closed form using certain approximations for the hydrodynamic interaction

and for the 2-body propagator. We find that memory effects convert the exponential decay of

density correlation functions to long time power law decay of the form t “*. We consider the

so-called long time self-diffusion constant and show that memory effects to first order in density

are negligible compared to the first cumulant contribution. Our treatment shows that for hard

sphere systems it is essential to treat the short distance hydrodynamic forces accurately.

1. Introduction

In a preceding article (I) we used a projection formalism to derive a low

density long wavelength expression for the memory matrix associated with

diffusion in a polydisperse suspension. In the present article we will explicitly

evaluate the memory matrix for hard spheres using an approximation for the

2-body diffusion propagator. The approximation is to neglect hydrodynamic

interaction in the propagator but to keep the hydrodynamic factors which

multiply the propagator. We find that the 2-body diffusion propagator

generates a branch point in the Laplace transform of the memory matrix

which implies a KS” long time power law decay of the density-density

correlation matrix. Our polydisperse formalism includes the case of self-

diffusion. We find that for the so-called long time self-diffusion constant,

memory effects make only a small alteration to the first cumulant result. We

show that other calculations of this long time result for hard spheres have not

correctly treated the short distance hydrodynamic forces. While our treatment

is an improvement on earlier results it too has short distance errors that can

be eliminated only by numerical study of the short distance hydrodynamic

forces.

0378-4371/82/OOOMIOOO/$O2.75 @ 1982 North-Holland

578 R.B. JONES AND G.S. BURFIELD

2. Direct force effects in the hard sphere limit

In (I) we derived a memory equation for the density-density autocor-

relation matrix F(k, t) of the form

dF(k, t) -- = - K(k)O(k)F(k, t) +

dr I M(k. t - t’)O(k)F(k, t’) dt’. (2.1)

In Laplace transform language this becomes

fi(k, s) = [sI + K(k)O(k)- h?(k. s)O(k)lm’O !(k). (2.2)

where F, K. 0, M. 1 are matrices in species space with I a unit matrix. In (I)

we also obtained the forms of K, 0 and M in the long wavelength limit to

order k’ and to first order in density. For the memory matrix we had

tiao(k, s) = k . fi”““(s) + k + B(k’), (2.3)

where

/L+“(s) = 2 n” dl e d”W/k~T~nt~(~) dl’G”“(I, 1’: s)h”‘(l’), L’IU

(2.4)

and for (Y f CT.

/Q”“(s): (n”n~r)‘/? I dl e P’(llikl~Thmr( _ 1) dl’G”“(l, 1’; s)h”“(lf). (2.5)

In these equations Greek indices are species labels and rr’l denotes the

number density of particles of species p in the suspension. The quantity h””

was given as follows:

h”“(ri” - r,J = - (kBTlm’Fi,,.iC, * (&!I + A”“(r),, - rtc,) - B”“(r,,, ~ r,J)

- V,,, * (A”“tr,,, - r,,,) ~~ B”“(r,,, - r,,,)). (2.6)

Here F’i,,,, = - Vi,4”“(Ir,_ - rlR/) is the direct force on particle i,, due to particle

i, and A, 6 are hydrodynamic interaction tensors’).

For hard spheres the direct potential $““(I) is an infinite step at 111 =

R, + R,, where R, denotes the radius of the sphere of species CY. We can

represent the hard sphere as the limit of a sequence of short range smooth

repulsive potentials which become infinite when the spheres touch. For such a

smooth potential we write the combination - (keT)m’Fim,iu exp( - 4”“/kRT) which occurs in eqs. (2.4) and (2.5) as Vim exp( - $““/k,T). If we now take the

hard sphere limit the function exp( - $“‘/kaT) becomes a step function and we

have a delta function from its derivative

MEMORY EFFECTS IN 'THE DIFFUSION OF A SUSPENSION II 519

Vi.&-d”“iw) = H 8((ri_ - r,“l- R, - R,). n ”

(2.7)

In the I integration (I = ri, - ri,,) the delta function contributes only when

spheres i, and i, touch. Because Fia,iP multiplies a hydrodynamic mobility

tensor in (2.6) we will obtain from the radial integration over I the result

I - (Do” 1 + A”“(l) - B”“(l)) evaluated at III = R, + R,. When the spheres touch

however, the series expansions*) of A, 6 break down and we must use exact

results for the total mobility tensor such as those given by Batchelor’). For

two spheres of identical radius R a comparison of A, B as defined in

reference I with Batchelor’s mobility tensor notation shows that

1 - (Do7 + A(l) - B(l)) = 6TrlR keT (A,,(;) - An(;))6 (2.8)

where A,, -A,? is that combination of mobilities in Batchelor’s notation

appropriate to the case when two spheres are pulled towards each other. For

that configuration A,, - An vanishes exactly when the spheres touch. An

equivalent statement is that the hydrodynamic friction force becomes infinite

when the spheres touch. This remains true in creeping motion hydrodynamics

even if we use slip boundary conditions4) and should be true also for spheres

of different sizes. Thus for hard spheres the direct force term in (2.6)

contributes nothing to memory effects at low densities. (Note that the cross

term in eq. (2.4) where Fi,,i” occurs in h,“(l’) is handled in the same way by

using the reciprocity relation for the Green’s function derived in (I) to replace

exp(- 4”“(l)/keT) by exp(- 4”“(l’)/kBT).) Even if F is not strict hard sphere

in character but is still very short range, say less than a particle radius, then

the exact mobility tensors will still be small and there will be little con-

tribution to memory effects from the direct force. For long ranged forces such

as screened Coulomb forces the direct forces would be important but we do

not consider this case further in this article.

From the above discussion we see that for polydisperse hard spheres it is

the divergence term in eq. (2.6) which produces memory effects to first order

in density’). In (I) we gave a general expression derived by a method of

reflections for these divergences valid for any hydrodynamic boundary con-

dition at the particle surfaces. For the rest of this article we specialize to the

case of strict hard spheres with stick boundary conditions. In that case we can

use reference 2 to get the following explicit form of h”“(I)

(2.9)

where r) is the solvent shear viscosity and I = ri, - ri,. The constants X”“, YaV

5 x0 R.R. JONES AND G.S. BURFIELD

and Z”” are

X ‘“” = 5 R ;..

(2. IO)

Because the expression (2.9) is derived by a series expansion in reflections it

is quite accurate at reasonable particle separations when the convergence of

the series is rapid. It is certainly not quantitatively accurate when the

separation of particle centres is less than 2(R,, + R,.). In the last section of this

article we will use numerical results given by Batchelor’) to estimate how well

h”“(l) is represented by eq. (2.9).

3. Memory matrix evaluation

To compute the integrals in eqs. (2.4) and (2.5) requires an explicit form of

the Green’s function G”“(I, I’; s). This was defined in (I) as the exact pro-

pagator for the relative diffusion of two particles including both direct and

hydrodynamic interactions. Unfortunately there is no way to obtain G”’

analytically if hydrodynamic interactions are included. For that reason we

approximate G”” by neglecting the hydrodynamic tensors A and B and

retaining only the free diffusion term (D’;, + II;;) 7. In appendix A we indicate

how a closed form expression for G”” may be obtained with this assumption.

This approximation is used in other comparable calculationsh.7.x). In appendix

B we investigate the type of errors involved and argue that short distance

errors are more significant than long distance errors and moreover that the

short distance error in G”” partly compensates for the short distance errors in

h”‘(l) noted in the previous section.

To evaluate the diagonal element 1\;[““(k. s) we require the following

integral:

k?d”“( s) _ II

dl dl’e d”L”“‘hH’(k . h”“(l))G”“(I. I’: .S )(k . h”“( I’)). (3. I)

With the approximate form of G”” from appendix A we can integrate over the

directions of I and I’ in (3.1) to obtain

x I

dl’Ac;“(l, 1’; s) F+F+- , I 1

(3.2)

MEMORY EFFECTS IN THE DIFFUSION OF A SUSPENSION II 581

where

3 AY(L 1’; s) = 2azD,,(D,y/s)m [ i,(v)kr(v’) - k;(2)T) m k,(v)k,(u’)]. (3.3)

Here D,,. = D{ + Dt;, R,, = R, + R,, v = (s/D,,)“*/, V’ = (s/D~~)“*!‘, z)T =

(s/D,,)“*R,,, and i,(v), k,(u) are related to modified Bessel functions as

explained in appendix A.

For the off-diagonal element l\;l”“(k, s), (Y f U, we require the integral

k2euu( s) = dl dl’ emm”“c’)‘kBT(k * h”(- I))G”“(I, I’; s)(k . h*“(Z’)).

After the integration over angles this becomes

es”(s) = - (F,’ j dl [y+y+y]

R a” cc

X I dl’AT”(1, 1’; s) T+F+$ 1 .

R (IO

Note that ear(s) = eUa(s). The memory matrix (2.3) becomes

n;i’+(k, s) = k* 2 n”dpLY(s), (3.6a)

and for p f y

$fwy(k, s) = k2(npnY)“2epr(s).

(3.4)

(3.5)

(3.6b)

The remaining radial integrations cannot be expressed in useful closed form

except in certain limits such as the small s (long time) limit. It is clear

however that both d’“(s) and e&“(s) are analytic functions of s apart from

square root branch points at s = 0, s = to. We define a principal branch for

these functions by taking a cut to run from s = 0 to s = - M along the negative

real axis of s. The autocorrelation matrix E(k, s) in eq. (2.2) also has these

branchpoints. For later reference we note that these branch-points arise from

the diffusion propagator G”“(I, I’; s) which is approximate. We hope that the

neglected hydrodynamic interactions will not alter the branchpoint structure

on which the long time results depend.

4. Long time behaviour

From eq. (2.2) we see that to zero order in density when interactions are

ignored the autocorrelation matrix E@(k, s) is diagonal of form (s +

582 R.B. JONES AND G.S. BURFIE1.D

k2Dz))‘G”P. In the time variable F”“(k, t) would be 6”” exp( -k’D{t) and the

autocorrelation function of scattered light as given in (I) would be a sum of

exponentials weighted by the scattering amplitudes of the different species.

The first order interaction effects change p’@(k, s) from a function with

simple poles on the negative real s axis to a function with a cut instead. The

matrix 6’@(k. s) now has a discontinuity across the cut for s real and

negative,

disc paO(k, s) = PnC((k, s + it)) - P”a(k. s - i0). (4. I)

which we will call the spectral function. The

reduce to a contour integral around the cut

integral over the spectral function,

F”“(k,t)=& 1 e,” disc fi‘“@(k, s) ds.

0

inverse Laplace transform will

which can be expressed as an

(4.2)

The poles in the zero order p@(k, s) correspond to delta functions in the

spectral function. The first order interaction effects spread out these delta

functions moving the poles through the cut just off the principal sheet of

p@(k, s). This means that the spectral function is still sharply peaked near the

free diffusion points s = -k2D$ but there is now spectral strength extending

to s = 0. At short times the peaks dominate to give exponential like decay.

But for t ti (k’D,$ only the spectral weight near s = 0 matters and this we

can obtain in closed form from the approximate expressions in section 3.

For simplicity we now assume that there are only two species of particle

present labelled a and b with number densities n”, nh. Using the results of (I)

for the matrices K and 0 we have that

(4.3)

where

u (I0 = k?{D;;[n”(C”” + c$“)+ nhC’kh] - nhdah(s)},

U uh = k2(n”nh)“2[D;Cah ~ @‘(s)l,

U ba = k2(n”nh)“2[D1;Cha ~ rh”(s)], (4.4)

U hb = k2{D;[nb(Chb + C'ib)+ n"chq] - n”d’“(s)}.

Here (7’“” = Cc + Cg+ CE”+ CT’ with the constants Cyi;;, C’E’. C‘S”. Cyj’, CT’

given elsewhere’). If we compute the inverse of (4.3) and use it in eq. (2.2)

keeping only terms up to first order in density we get

MEMORY EFFECTS IN THE DIFFUSION OF A SUSPENSION II 583

l+p(k, s) = (s + k’Do”)-‘[Sdp - (n"nP)"*C~] -(s + k*D;)-‘(s + k*D{)-'uaP.

(4.5)

The branchpoint occurs here in the second term of (4.5) so that

nh disc d”“(s)

disc F(k, s) = k* (s + k*D;;)*

(n“n”)“* disc eho(s) (s + k*D;f)(s + k*D;)

(non’)“* disc e”‘(s) (s + k*D$)(s + k*D$j

n” disc d’“(s) (4.6)

(s + k*Dt;)’

To obtain the discontinuities of d”“(s), e@(s) for small s we use the result

disc AyP(l, 1’; s) =

(4.7)

for s real and negative. With this small s expression the remaining radial

integrals in (3.2) and (3.5) can be carried out to give the following expressions:

27 Xa”Z”p 25 (Y”p)2 5 Y”pz”” 27 (Z+‘)’

‘224 RS,, $864 Rt8 +112 R& +i%iT (4.8)

disc e@(s) = 2rD (b&(+jy~X;~~~ I ;:W”Y”~~J”“Y”“) aB

27 (X”“Zap + X@Z@“) 25 Y ““Y ap %224 R& +m R&

5 (Y”“z”p+ Y”PZ”“) 27 Z”“Z”” $112 R7,, +784 R$ I ’ (4.9)

To simplify these complex expressions let us henceforth assume that

species a and species h are identical apart from light scattering power. This

assumption will also facilitate discussion of self-diffusion when we can take

species a to be a single tagged particle among otherwise identical h type

particles. Under these conditions R, = Rb = R, 0: = 0; = Do and d”‘(s) =

d’“(s) = -e”‘(s). We can then obtain to lowest order in s

disc 6’(k, s) = $ (&O)($-)3”(&$)2~ ( _(n?ib)l,T -(numb)“‘).

(4. IO)

Here 1.9 is a numerical factor arising from evaluation of (4.8) for identical

particles. If we use this result in eq. (4.2) we obtain as the leading term at long

5x4 R.B. JONES AND G.S. BURFIELD

times

(4.11)

where 4” is a volume fraction, 4” = 47rR’n”/3.

We have shown that the diffusion induced branchpoint at s = 0 leads to a

t-“’ decay for F(k. t) at long times rather than pure exponential decay. This

weak long time tail may not be accessible to experimental observation

however because by the time it becomes dominant over the initial exponential

decay very many particle collisions will have occurred so that contributions

of higher order in density may be significant as well. Although our calculation

shows M(k. t) and F(k, t) to be long lived it is common in the literature to talk

about a long time diffusion constant obtained by decoupling the memory term

in eq. (2.1) by writing at large t

dF(k, t) ~ = ._ dt KOF(k. r)+ (1 M(k. t’) dt’)OF(k, t)

= -(K - ti(k. O))OF(k, f)

= ~ K,~OF(k. t). (4.12)

where Kr. is a long time cumulant matrix. We can readily evaluate K, in our

calculation. We use the result

to obtain

(4.13)

(4.14)

39 ZO”Z”” + 1078 R::

(4.15)

To simplify matters we again take species a and h to be-identical as in the

derivation of (4.1 I). One then obtains

MEMORY EFFECTS IN THE DIFFUSION OF A SUSPENSION II 585

ti(k, 0) = 0.08k2Do(_ ($$,),,2 -(‘;$b)“2),

which together with K from (I) gives an explicit result for KL.

(4.16)

5. Self-diffusion

We get the autocorrelation function of self-diffusion in our formalism by

choosing species a to consist of one particle only while species b comprises

all other particles which can be taken as identical in structure to particle a.

Then eqs. (4.11) and (4.16) directly give self-diffusion results by setting

4“ = 0, $b = 4. From eq. (4.11) we predict for self-diffusion a long time tail of

the form

FS(k, t) = F”“(k, t) = 21.2y (A) 5/z

(5.1)

If we try to identify a long time diffusion constant from eqs. (4.12) and (4.16)

we obtain at large t

dFS(k t) 1- = - k2D&FS( k, t) = - k2Do[ 1 + nCA - 0.08+]Fs(k, t).

dt

The term nCA arises from the first cumulant. Using the expression for A calculated by reflections Felderhof’) obtained nCA = - 1.734 while a numeri-

cal evaluation by Batchelor3) gave nCA = - 1.834. Since we are using the

tensors A and 6 as calculated by reflections we will use Felderhof’s result.

The memory term contributes the very small correction -0.084 to the first

cumulant result. The net result to first order in volume fraction is then

D&, = Do(l - 1.814). (5.2)

We can use (5.1) to see that the long time tail is difficult to detect

experimentally. Assuming 4 = 0.1 and kR = 0.1 we compare the magnitude of

(5.1) with the exponential result for non-interacting diffusion exp(- k2Dot).

The tail in (5.1) becomes comparable to or greater in magnitude than the

exponential only for times t such that k2D t 0 b 20. This time scale is too long

for present experimental techniques6). Moreover on such a time scale the

particle would diffuse a distance of about a hundred interparticle separations

and thus could have collided many times with other particles. In such a time

regime a result calculated only to first order in density is of doubtful

relevance to any experimental observations.

The result (5.2) for D&,,,, is noteworthy chiefly because it is essentially

unchanged from the first cumulant result. In other calculations much larger

586 R.B. JONES AND G.S. BURFIE1.D

memory effects are quoted. For example one evaluation”) of a mode coupling

expression to first order in density gave

where the -44/j represents the mode coupling or memory effect. A sub-

sequent calculation by Marqusee and Deutchx) criticised this result on the

basis that Oseen tensor contributions to the hydrodynamic interaction

changed - 4&/3 to - 0.074. On the basis of the calculations presented here we

can say that the factor -4/3 corresponds to keeping the free diffusion part of

(2.6), which is DEl, and putting the tensors A and B equal to zero while

approximating the 2-body propagator by a free diffusion propagator without

any excluded volume effects (keeping only the 1/(47~D,,~~) in (4.13)). The

correction to this which Marqusee and Deutch*) calculate keeps the Oseen

part of B in the direct force part of (2.6) but neglects A and the divergence

terms completely. What we have shown is that the direct force contribution is

identically zero to first order in density and the entire first order result of

-0.084 is from the divergence terms of (2.6). Our conclusion is that to first

order in density DF,,,ny is dominated completely by the first cumulant effect of

the static background of other particles mediated by the tensor A which is the

source of the - 1.73b term.

6. Discussion

Our results in the preceding article (I) and in the present article indicate that

for hard sphere suspensions or for suspensions with very short range direct

forces it is essential to treat the short range hydrodynamic interaction

accurately. By use of series expansions of the mobility tensors we have

improved on previous calculations at low densities but our results also

incorporate approximations. The chief qualitative result of our calculation is

that at long times memory effects change the exponential decay of density

correlations into power law decay as t “‘. In a calculation for a Coulomb

suspension Hess and Klein’) find a similar t mvz tail for the velocity autocor-

relation function of self-diffusion. In both cases the long tail arises from the

use of an approximate free diffusion propagator without hydrodynamic inter-

action in evaluating memory terms like m@(s). In appendix B we show that at

large distances this approximation is reasonable although it fails at short

distances. We expect that the t ~“’ behaviour will survive in an exact treat-

ment of the propagator but this point merits further study.

One quantitative result of our work is that, for the long time self-diffusion

MEMORY EFFECTS IN THE DIFFUSION OF A SUSPENSION II S87

constant, memory effects to first order in density are small as compared to the

first cumulant contribution. The numerical factor - 0.08 in eq. (4.16) is not to

be believed precisely but we think it is correct in order of magnitude. There

are two sources of error in obtaining this small number. One is that the

expression (2.9) for the divergence of the relative diffusion tensor is only

approximate. As explained in appendix B we can use numerical’) and exact4)

results for two identical hard spheres to obtain more precise values for this

divergence term at short distances. When the spheres touch our result in (2.9)

is a factor of 3.5 smaller than the exact result. When the sphere centres are

separated by three radii our expression is wrong by only about one per cent

and at greater separations our expression should become very accurate. Since

h”O occurs quadratically in the memory function we appear to have made a

serious underestimate of short distance contributions to the memory term.

This error however is partly compensated by the approximate Green’s func-

tion given in appendix A. The exact propagator should in fact vanish when the

spheres touch as argued in appendix B. Therefore although our ha@ is too

small at short distances, our Green’s function is too large and thus we expect

partial compensation of the two errors.

We conclude that for diffusion of dilute hard spheres the first cumulant

result should be reasonable even at experimentally long times of several

collision times. At higher densities many body effects will be more

pronounced and the present calculation suggests that in attempting to cal-

culate these great care will be needed with the short distance hydrodynamic

forces. In contrast, for suspensions with long range direct forces, the

dominant contribution to haP will be from the direct force term in eq. (2.6).

The short range behaviour of propagators and mobilities will not be so

important and it may well be a good approximation to use free diffusion

propagators and to approximate A and 6 by the first few terms from the

series of reflections.

Appendix A

The exact propagator of relative diffusion for two particles of species p and

v, G’““(I, I’; s), was defined in (I) as the solution to the differential equation

[s _ eP++“(I)v . e-P4JWl) T’“(1). V]G’“(l, I’; s) = 6(1 - I’), (A.1)

where p = (keT)-‘,

T”“(I) = (06 + Do”) 1 + AYl) + A”“(I) - 26’7l), (A.2)

and I = rF - rv. The boundary condition is that the radial component of VGP”

5X8 R.B. JONES AND G.S. HURFIEI>D

vanishes at infinite particle separation and also when the particle hard cores

touch. A widely used approximation7.X) consists of dropping the A, I3 tensors

while keeping the free diffusion part of T”“. Making this approximation for

hard spheres means we must solve the simpler equation

[s - D,,O’]GYl. I’; .s) = 6(1 ~~ I’). (A.3)

subject to the boundary condition aG’“/JI = 0 at 1 = R,,, -= R,, + R,. and at

I = r. Eq. (A.3) is readily solved by expansion in Legendre functions of cos 8.

where 8 is the angle between 1 and I’. We write

G*“ll, 1’; s) = 2 A$“(/, I’; .s)P,,(cos 0). n -0

Inserting this into eq. (A.3) we get for 1* 1’

Defining I’ = (s/D,,.)“‘l, the independent solutions of (A.5) are

where I,, tl~~~h K n t,l,Z, are modified Bessel functions.

For A:“(/, 1’: s) we write

A:“(I, I'; s) = u,&(E)+ h,,k,(r). I < I’

c,,,k,,(L.). I I I’.

(A.4)

(A.5)

(A.6)

(A.7)

By integrating (A.3) over an infinitesimal volume about I = I’ in the standard

way’“) we obtain two conditions at I = I’, firstly a continuity condition.

c‘,,k,, - u,,i,, - h,,k,, = 0. (A.8)

and secondly a jump condition on the first derivatives.

When the spheres touch at I = R,,. we have the boundary condition

” dl di, + h, !!!$ = 0,

One can readily solve these three conditions to obtain for I < I’.

(A.9)

(A. IO)

MEMORY EFFECTS IN THE DIFFUSION OF A SUSPENSION II 589

where UT = (s/D,,)“‘R,+ and i;, k; are first derivatives of i,, k,. For hard

spheres we have from the reciprocity condition given in the appendix to (I)

that .4:“(1, 1’; s) = A:“(I’, 1; s) which completes the definition of A:“(I, 1’; s).

Appendix B

Here we consider further the exact 2-body propagator for hard spheres

which is a solution to eq. (A.1). For simplicity let us assume two identical

spheres of radius R. The Green’s function equation becomes

[S - (V,Tij(L))Vi - Tij(l)ViVj]G(1, 1’; S) = 6(1 - I’), (B.1)

with

T(I) = 2[Do 1 + A( I) - i3( i)]. (B.2)

We may write T(I) in a way which relates to Batchelor’s notation3) as

T(1) = ${a(f)$+b(;)(l-$)),

with

a(f)=&($)-An($),

b($)=&,(f)-h(;).

(B.3)

(B.4)

Here AlI - AI2 and B,, - B12 are combinations of mobilities given numerically

by Batchelor3). One now can easily check that the divergence of T can be

expressed as

V . T(I) = I

f. (B.5)

Using spherical polar coordinates with I’ as polar axis one expands G in

Legendre functions as in (A.4) and obtains a differential equation for

A,(1, 1’; s) which for If 1’ is

u3.6)

with the boundary conditions dA,ldl = 0 at I = 2R and 1 = 00.

The exact short distance behaviour of a(l/R) has been given by Wolynes

and Deutch4) based on the exact solution of the creeping motion hydro-

dynamic equations in bispherical coordinates due to Brenner”). As 1+ 2R one

590 R.B. JONES AND G.S. BURFIELD

has

a(l/R) = 2(1/R - 2). (8.7)

Using (B.7) together with Batchelor’s tables’) of u(l/R) and h(l/R) enables us

to calculate V * T with reasonable precision for 1 = 2R (spheres touching) and

I = 3R. It is these values that were compared with the results from the

expansion in reflections in section 6. Since u(l/R) vanishes while h(l/R) tends

to a constant when the spheres touch we see that the differential equation

(B.6) forces A,,(!, 1’; s) to vanish as I -+ 2R. Thus the exact Green’s function

vanishes when the spheres touch unlike the approximate Green’s function in

appendix A. In the opposite extreme of large separation one has that4) as

I -+m, a(l/R) = 1 + O(R/I) and also b(l/R) = I + O(R/I) so that eq. (B.6) reduces

in that limit to the approximate equation (A.5). It would seem then that at

large distances the hydrodynamic effects are small corrections to the free

diffusion approximation while at short distances the corrections are large. It is

plausible that the solution to (B.6) still retains the branch point at s = 0 but we

do not know enough about the analytic properties of the functions ~r(l/R),

b(l/R) to prove this yet.

References

I) R.B. Jones, Physica 97A (1979) 113: 1llA (1982) 262. to be referred to as paper (1).

2) P. Reuland, B.U. Felderhof and R.B. Jones. Physica 93A (197X) 465.

3) G.K. Batchelor. J. Fluid Mech. 74 (1976) I.

4) Peter G. Wolynes and J.M. Deutch. J. Chem. Phys. 65 (1976) 450.

5) The fact that the non-vanishing divergence of the relative mobility tensor is significant in a

polydisperse suspension was first pointed out to our knowledge by Prof. G.K. Batchelor in an

invited lecture at the meeting on Structure and Dynamics in Dispersions of Interacting

Particles held at Leeds University in July 1980.

6) P.N. Pusey and R.J.A. Tough, in Dynamic I,ight Scattering and Velocimetry: Applications of

Photon Correlation Spectroscopy, R. Pecora. ed. (Plenum. New York. 19x1).

7) W. Hess and R. Klein. Physica 105A (1981) 552.

8) J.A. Marqusee and J.M. Deutch, J. Chem. Phys. 73 (1980) 5396.

9) B.U. Felderhof. J. Phys. All (1978) 929.

10) Jon Mathews and R.L. Walker, Mathematical Methods of Physics 2nd ed. (Benjamin, New

York, 1970).

II) Howard Brenner. Chem. Eng. Sci. 16 (1961) 242.